
Various forms of the superoperator \(\mathcal {L}\)  can be derived especially in the regime where
the coupling between the system and the bath is weak.
The Redfield equation can either be obtained
by applying the Born-Markov approximation or by using the time-convolutionless projector operator formalism [1]}, [2]}.
In the interaction picture, denoted by tilde as \(\tilde{\rho }(t) = e^{i H_{\rm S} t} \rho (t) e^{-i H_{\rm S} t} \)  and \(\tilde{H}_{\rm I} (t) = e^{i H_{\rm S} t} H_{\rm I} e^{-i H_{\rm S} t} \) , it is given by
\(\frac{d\tilde{\rho }(t)}{dt}=-\int _0^t ds\; \mathrm {Tr}_{\rm B} [ \tilde{H}_{\rm I} (t),[\tilde{H}_I(t-s),\tilde{\rho }(t)\otimes \rho _{\rm B}]].\) 
